Use Pythagorean identities to prove whether ΔLMN is a right, acute, or obtuse triangle. Show all work for full credit.

 The given triangle LMN is an obtuse-angled triangle.

Step-by-step explanation:  We are given to use Pythagorean identities to prove whether ΔLMN is a right, acute, or obtuse triangle.

From the figure, we note that

in ΔLMN, LM = 5 units, MN = 13 units  and  LN = 14 units.

We know that a triangle with sides a units, b units and c units (a  > b, c) is said to be

(i) Right-angled triangle if

(ii) Acute-angled triangle if

(iii) Obtuse-angled triangle if

For the given triangle LMN, we have

a = 14, b = 13 and c = 5.

So,

Therefore,  

Thus, the given triangle LMN is an obtuse-angled triangle.